Raising and Lowering operators of spin-weighted spheroidal harmonics

In this paper we generalize the spin-raising and lowering operators of spin-weighted spherical harmonics to linear-in-$\gamma$ spin-weighted spheroidal harmonics where $\gamma$ is an additional parameter present in the second order ordinary differential equation governing these harmonics. One can then generalize these operators to higher powers in $\gamma$. Constructing these operators required calculating the $\ell$-, $s$- and $m$-raising and lowering operators (and various combinations of them) of spin-weighted spherical harmonics which have been calculated and shown explicitly in this paper.


I. INTRODUCTION
Spin-weighted spheroidal harmonics, S γ ℓ,m s , arise naturally in any analysis of the angular dependence of propagating fields on rotating, Kerr black hole space-time backgrounds 1,2 , and are most studied in the differential equations governing linear electromagnetic and gravitational perturbations. When the spin, s, of the propagating field is zero, these angular eigenfunctions become the oblate (scalar) spheroidal harmonics 3 . When the black hole is spherically symmetric, the full angular eigenfunctions are the spin-weighted spherical harmonics, well known in other areas of Physics. In this work we will focus, for the first time, on describing the operators for raising and lowering the spin index of the spin-weighted spheroidal harmonics. We will do this to first order in the parameter perturbing away from the spin-weighted spherical harmonics, and lay the groundwork for extending the procedure to higher order. For simplicity, we will generally assume an unwritten factor of e imφ throughout, and shall concentrate primarily on the θ-dependence, since the azimuthal eigen-equation is rather trivial.
Spin-weighted spheroidal harmonics satisfy the angular part of Teukolsky's master equation: where s is the spin weight of the harmonic, and s E γ ℓ,m is the eigenvalue which, in the limit γ → 0, is ℓ(ℓ + 1). As with any 2 nd order differential equation, Eq.
(1) has two linearly independent solutions, one of which, s S γ ℓ,m is generally used for describing scalar, (massless) neutrino, electromagnetic and gravitational perturbations. In the limit γ → 0, these harmonics are the spin-weighted spherical harmonics, s Y ℓ,m . In the limit s → 0, s S γ ℓ,m are the ordinary spheroidal harmonics, S γ ℓ,m , and s Y ℓ,m are the ordinary spherical harmonics, Y ℓ,m . The s Y ℓ,m appear as a) Electronic mail: a.g.shah@soton.ac.uk b) Electronic mail: bernard@phys.ufl.edu a solution to the equation To build s Y ℓ,m , one repeatedly applies spin raising and lowering operators on ordinary spherical harmonics, separately computing eigenfunctions with positive and negative values of spin-weight: where Here ð s is the raising operator,ð s is the lowering operator, and P m ℓ are the associated Legendre functions. ð and ð are given a subscript here to show which spin-weighted quantities they act on. For each s, the s Y ℓ,m are complete and orthogonal functions on the 2-sphere, and are related to the Wigner D-rotation matrices by Unlike the way s Y ℓ,m are calculated using raising and lowering operators, spin-weighted spheroidal harmonics are usually calculated as a sum over s Y ℓ,m 4 or as a sum over Jacobi polynomials 5 . In this paper we work on generalizing ð s andð s to operators that raise and lower the spin-weight of spheroidal harmonics, s S γ ℓ,m . That is, z = s S γ ℓ,m and y = s±1 S γ ℓ,m are solutions to two different differential versions of Eq. (1), one being sˆ γ ℓ,m z = 0 and the other being s±1ˆ γ ℓ,m y = 0, and we will find a relation of the form: to linear order in γ, between the solutions (y and z) of these equations. The paper is organized as follows. In Section II, we summarize Whiting's earlier work 6 on finding relations between solutions of two differential equations. In Section III, we use this work to calculate the different ℓ-, sand m-raising and lowering relations of s Y ℓ,m . In Section IV, we build the linear-in-γ s-raising and lowering operators for s S γ ℓ,m .

II. EARLIER WORK ON RELATING SOLUTIONS OF TWO DIFFERENTIAL EQUATIONS
Relations of the general form which we seek have been studied previously by one of us 6 and were extensively used in 7 to show mode stability for the perturbations being discussed here. We now give a brief, and slightly more general, introduction, while more complete details can be found in 6 . Thus, we suppose that y(x) and z(x) satisfy y ′′ + py ′ + qy = 0 and z ′′ + P z ′ + Qz = 0, in which ′ = d/dx, and seek conditions that α and β must satisfy in order that should hold. More specifically, since each of Eq. (7) is second order, two linearly independent solutions exist, say (y 1 , y 2 ) and (z 1 , z 2 ) respectively, and we will actually demand that the mapping (8) applies more fully, so that: That is, every solution for y will map to a solution for z. Defining the relevant Wronskians by: where C y and C z are constants, we can invert Eqs (9) to find α and β: Clearly, α and β are determined entirely by the solutions they map between. Differentiating (8) once and using Eq. (7) for z we find: Eqs (8) and (12) together can be inverted to give z and z ′ in terms of y and y ′ . For this we will also need to define: Then Further differentiating Eq. (12), and using both Eqs (7), we can deduce: in which each coefficient must separately be zero because of Eqs (9). Thus: With the appropriate combination of these, we can now show constructively that: as already follows from Eqs (10) and (13) above. In the application we have in mind, P = p, so that k = const.
We could also check the integrability of Eqs (14) which, with Eq. (17) and some algebra, yields the second of Eqs (16). Finally we note that the operators in Eq. (7) can be written as: in which the first order operators are effectively intertwined.

III. SPIN-WEIGHTED SPHERICAL HARMONICS
Let us denote spin-weighted spherical harmonics of type-1 and type-2 by s Y ℓ,m and s X ℓ,m , being two linearly independent solutions of Eq (2), where, in the notation of section II, To build the harmonics of non-zero spin, we begin with spin-weight zero ordinary spherical harmonics (suppressing e imφ ): and apply further s-raising and s-lowering operators to generate arbitrary spin-weighted spherical harmonics: Therefore, and same holds for s X ℓ,m . From Section II we know that to find relations between solutions of equations z ′′ + P z ′ + Qz = 0, y ′′ + py ′ + qy = 0 (23) we need to calculate where W (z 1 , z 2 ) is defined in (10). One then has Finding β's and α's for various relations, we get The above six relations are equivalent to Gauss's relations for contiguous functions of the hypergeometric function, 2 F 1 (a, b, c; z). By equating ∂ θ s Y ℓ,m of two different relations, one can get various numbers of recurrence relations. For example, by equating ∂ θ s Y ℓ,m in Eqs (26) and (29), one gets a relations between s Y ℓ,m , s+1 Y ℓ,m and s Y ℓ−1,m . By repeated application of this procedure, various relations can be formed between different s+i Y ℓ+j,m+k (where i, j, k are integers). Finally, as an example of Eq. (18) we show:
In general cases, it is straight forward to use , and (A2) where | are the Clebsch-Gordan coefficients which can be calculated using Now, we study the following special cases.